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Assume a labor-intensive production function:

$Q(L)=L^\beta$

Find demand of labor that maximizes profits (unconditional demand $L(p,w,r)$), the demand of labor that minimizes costs (unconditional demand $L(q,w,r))$, the supply function $(Q(L(p,w,r))$), and the benefits $(\pi(L(p,w,r))$ if:

a) $\beta>1$

b) $\beta=1$

For the first case (increasing returns to scale) I am able to Maximize profits by getting the First Order Conditions, etc., and finding $L(p,w,r))$:

$L(p,w,r))=(w/p\beta)^{1/\beta-1}$

Thus, by replacing $L(p,w,r))$ into the benefits:

$\pi(L(p,w,r)=p*(w/p\beta)^{\beta/\beta-1}-w*(w/p\beta)^{1/\beta-1}$ (It can be simplified by doing more algebra).

By replacing $L(p,w,r))$ into the production function:

$Q(L(p,w,r)=(w/p\beta)^{\beta/\beta-1}$

Finally, the conditioned (on a production of "q") labor demand is given by:

$L(q,w,r)=q^{1/\beta}$

However, for the second case (constant returns to scale) I am not able to get the First Order Conditions and so on for obtaining the benefits, the supply function, etc. Thus, I would like to know how to address this problem.

I would appreciate any help,

Thanks in advance

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  • $\begingroup$ You'll probably have to consider corner solutions for the case where $\beta=1$. Also, why is labor demand a function of $r$ (presumably rent) when capital is not required in production? $\endgroup$ – Herr K. Feb 13 at 17:19
  • $\begingroup$ You're right! My bad, I just wrote the standard form but sure, it only depends on $w$ and $p$, in the case of non-conditioned labor demand. For a single-input production function, corner solutions just refer to the case when $L(p,w)=0$? $\endgroup$ – F. Sánchez Feb 13 at 19:58
  • $\begingroup$ Depending on the ratio of $p/w$, corner solution could be $0$ or the upper bound on labor supply. $\endgroup$ – Herr K. Feb 13 at 20:14

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